Indeed. It took me 4, and a couple of them are demanding, though interesting. I certainly won't claim that this is the most obvious route, and I expect that others will find less "extreme" paths. However, I believe that all paths are learning experiences.

There is a 18UR in the cells marked #. One way to view the extra digits in this UR is as the <4> in r1c2 having a strong inference with the 39 group of <9> in r1c3 together with the grouped <3>s of r9c23, or:
18UR[((3)r9c23(9)r1c3)=(4)r1c2]
The parentheses around "(3)r9c23(9)r1c3" indicate that it is a group. A group is true if any of its members is true; and it is false if all of its members are false. A fairly simple branched AIC can render all of the members of this group false, leading to the next elimination:

There is a 6-cell 12DP marked #. The extra digits provide the folowing induced strong inference between the grouped <4>s in r1c4 and r7c6 and the <5> in r7c6:
6Cell12DP[(4)r1c4|r7c6=(5)r7c6]
This provides the following AIC:
(4)r7c4 - 6Cell12DP[(4)r1c4|r7c6=(5)r7c6] - ALS[(5)r4c6=(3)r34c6] - (3=4)r8c6 - (4)r7c4; r7c4<>4
This solves the puzzle.

Somebody please show me that I am overlooking something obvious!

[Edit to repair the "broken" display of the AIC in "code".]
[Edit again to correct the XY Chain address per Keith's edit, as shown in red above.]

Last edited by Asellus on Fri Nov 28, 2008 4:52 am; edited 1 time in total

First, I'm interested in why you chose to make a loop out of the 4-cell XY-Chain.

I don't really understand your question. ALL XY Chains that perform eliminations are (discontinuous) loops. If there happen to be multiple victims, they are included at the discontinuity. It is still a loop, regardless. In general:
Victim A|Victim B|Victim C|etc - AIC - Victim A|Victim B|Victim C|etc; A,B and C, etc., are false

First, I'm interested in why you chose to make a loop out of the 4-cell XY-Chain.

I don't really understand your question. ALL XY Chains that perform eliminations are (discontinuous) loops. If there happen to be multiple victims, they are included at the discontinuity. It is still a loop, regardless. In general:
Victim A|Victim B|Victim C|etc - AIC - Victim A|Victim B|Victim C|etc; A,B and C, etc., are false

Interesting perspective.

To me, many chains are simply a forcing net where the first inference stream is assumed. XY-Chains are a prime example. Take the heart of your AIC loop for example:

Thus, any cells that are peer cells of [r2c9] and [r8c1] will have (9) eliminated.

I thought the point of AICs being bidirectional was to allow a similar conclusion. If we assume that [r2c9]<>9, then reading the AIC chain from left-to-right guarantees that [r8c1]=9 is true. Similarly, if we assume that [r8c1]<>9, then reading the AIC from right-to-left guarantees that [r2c9]=9 is true. No matter what, the AIC guarantees that at least one of [r2c9],[r8c1] is true for (9).

I thought the point of AICs being bidirectional was to allow an equivalent conclusion without resorting to forcing nets.

I can't imagine a more obvious AIC than an XY Chain. The strong inferences occur within the bivalue cells and the weak inferences occur between the bivalue cells. Alternation of inferences is assured. The pincer ends are each weakly linked to the victim(s), providing the weak link discontinuity for the closed loop. There is no need whatsoever to refer to forcing concepts. One only needs to note the sequence of inferences to perform the elimination(s). I can assure you that a forcing net never even entered my consciousness. (Why would one resort to forcing in the most obvious of all AICs unless one was, for some reason, reluctant to rely upon the inherent logic of AICs themselves?)

While the use of starting assumptions is, in an important sense, implicit in any AIC, the AIC structure obviates the need for any such explicit assumption (i.e. "forcing"). That is what makes it appealing. And, since almost any (if not every ... I await proof to the contrary) elimination or placement in sudoku can be expressed in terms of an AIC, forcing appears to be unnecessary.

[Edit: Somehow, between the time I quoted the previous message and the time I posted the response, the text I quoted had changed.]

I had another variation on the theme without the UR endeavor. I also started with the skyscraper on <4> and then I found the xy-chain noted by Asellus but viewed in reverse order. Then I spotted a finned xy-wing on <134> that deleted a <1> from r1c4, but I do not think it was helpful in the long run. I finished with three xy-wings, two again on <134> and then <124> with a (probably unhelpful) coloring on <1> somewhere in that mix.